Twodimensional variational problems with linear growth

Abstract

Suppose that f:\mathbb{R}^{nN}\rightarrow\mathbb{R} is a strictly convex energy density of linear growth, f(Z)=g(\left|Z\right|^{2}) if N>1. If f satisfies an ellipticity condition of the form D^{2}f(Z)(Y,Y)\geq c(1+\left|Z\right|^{2})^{-\frac{\mu}{2}}\left|Y\right|^{2},\;1<\mu\leq3, then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem \int_{\Omega}f(\nabla w)dx+\int_{\partial\Omega}f_{\infty}((u_{0}-w)\otimes v)d\mathcal{H}^{n-1}\rightarrow\mbox{min in }W_{1}^{1}(\Omega;\mathbb{R}^{N}), provided that the given boundary data u_{0}\in W_{1}^{1}(\Omega;\mathbb{R}^{N}) are additionally assumed to be of class L^{\infty}(\Omega;\mathbb{R}^{N}).Moreover, if \mu<3, then the boundedness of u_{0} yields local C^{1,\alpha}-regularity (and uniqueness up to a constant) of generalized minimizers of the problem \int_{\Omega}f(\nabla w)dx\rightarrow\mbox{min in }u_{0}+W_{1}^{1}(\Omega;\mathbb{R}^{N}). In our paper we show that the restriction u_{0}\in L^{\infty}(\Omega;\mathbb{R}^{N}) is superfluous in the twodimensional case n=2, hence we may prescribe boundary values from the energy class W_{1}^{1}(\Omega;\mathbb{R}^{N}) and still obtain the above results.